LMFDB - Embedded newform 4923.2.a.l.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.3103529151$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.523506$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.523506 q^{2} -1.72594 q^{4} +1.35183 q^{5} +3.60927 q^{7} -1.95055 q^{8} +O(q^{10})$	\(q+0.523506 q^{2} -1.72594 q^{4} +1.35183 q^{5} +3.60927 q^{7} -1.95055 q^{8} +0.707691 q^{10} +0.337880 q^{11} -0.968153 q^{13} +1.88948 q^{14} +2.43075 q^{16} -1.24740 q^{17} -5.50049 q^{19} -2.33318 q^{20} +0.176882 q^{22} -3.13569 q^{23} -3.17256 q^{25} -0.506834 q^{26} -6.22939 q^{28} +7.18364 q^{29} +10.2089 q^{31} +5.17362 q^{32} -0.653022 q^{34} +4.87912 q^{35} +5.45502 q^{37} -2.87954 q^{38} -2.63682 q^{40} +7.24577 q^{41} -6.48922 q^{43} -0.583161 q^{44} -1.64155 q^{46} +7.68104 q^{47} +6.02683 q^{49} -1.66085 q^{50} +1.67097 q^{52} +4.50651 q^{53} +0.456756 q^{55} -7.04008 q^{56} +3.76068 q^{58} -2.41312 q^{59} +6.37922 q^{61} +5.34441 q^{62} -2.15308 q^{64} -1.30878 q^{65} +0.799497 q^{67} +2.15294 q^{68} +2.55425 q^{70} -2.93852 q^{71} -15.3091 q^{73} +2.85574 q^{74} +9.49353 q^{76} +1.21950 q^{77} +10.3913 q^{79} +3.28597 q^{80} +3.79321 q^{82} +13.3213 q^{83} -1.68627 q^{85} -3.39715 q^{86} -0.659054 q^{88} +4.42608 q^{89} -3.49433 q^{91} +5.41201 q^{92} +4.02107 q^{94} -7.43573 q^{95} -6.98217 q^{97} +3.15508 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.523506	0.370175	0.185087	−	0.982722i	$-0.440743\pi$
$2$	0.523506	0.370175	0.185087	+	0.982722i	$0.440743\pi$
$3$	0	0
$3$	0	0
$4$	−1.72594	−0.862971
$5$	1.35183	0.604556	0.302278	−	0.953220i	$-0.402253\pi$
$5$	1.35183	0.604556	0.302278	+	0.953220i	$0.402253\pi$
$6$	0	0
$7$	3.60927	1.36418	0.682088	−	0.731270i	$-0.261072\pi$
$7$	3.60927	1.36418	0.682088	+	0.731270i	$0.261072\pi$
$8$	−1.95055	−0.689625
$9$	0	0
$10$	0.707691	0.223792
$11$	0.337880	0.101875	0.0509374	−	0.998702i	$-0.483779\pi$
$11$	0.337880	0.101875	0.0509374	+	0.998702i	$0.483779\pi$
$12$	0	0
$13$	−0.968153	−0.268517	−0.134259	−	0.990946i	$-0.542865\pi$
$13$	−0.968153	−0.268517	−0.134259	+	0.990946i	$0.542865\pi$
$14$	1.88948	0.504984
$15$	0	0
$16$	2.43075	0.607689
$17$	−1.24740	−0.302539	−0.151270	−	0.988493i	$-0.548336\pi$
$17$	−1.24740	−0.302539	−0.151270	+	0.988493i	$0.548336\pi$
$18$	0	0
$19$	−5.50049	−1.26190	−0.630950	−	0.775824i	$-0.717335\pi$
$19$	−5.50049	−1.26190	−0.630950	+	0.775824i	$0.717335\pi$
$20$	−2.33318	−0.521714
$21$	0	0
$22$	0.176882	0.0377115
$23$	−3.13569	−0.653836	−0.326918	−	0.945053i	$-0.606010\pi$
$23$	−3.13569	−0.653836	−0.326918	+	0.945053i	$0.606010\pi$
$24$	0	0
$25$	−3.17256	−0.634511
$26$	−0.506834	−0.0993984
$27$	0	0
$28$	−6.22939	−1.17724
$29$	7.18364	1.33397	0.666984	−	0.745072i	$-0.267585\pi$
$29$	7.18364	1.33397	0.666984	+	0.745072i	$0.267585\pi$
$30$	0	0
$31$	10.2089	1.83357	0.916784	−	0.399384i	$-0.130776\pi$
$31$	10.2089	1.83357	0.916784	+	0.399384i	$0.130776\pi$
$32$	5.17362	0.914576
$33$	0	0
$34$	−0.653022	−0.111992
$35$	4.87912	0.824721
$36$	0	0
$37$	5.45502	0.896799	0.448400	−	0.893833i	$-0.351994\pi$
$37$	5.45502	0.896799	0.448400	+	0.893833i	$0.351994\pi$
$38$	−2.87954	−0.467124
$39$	0	0
$40$	−2.63682	−0.416917
$41$	7.24577	1.13160	0.565800	−	0.824543i	$-0.308568\pi$
$41$	7.24577	1.13160	0.565800	+	0.824543i	$0.308568\pi$
$42$	0	0
$43$	−6.48922	−0.989597	−0.494798	−	0.869008i	$-0.664758\pi$
$43$	−6.48922	−0.989597	−0.494798	+	0.869008i	$0.664758\pi$
$44$	−0.583161	−0.0879149
$45$	0	0
$46$	−1.64155	−0.242034
$47$	7.68104	1.12039	0.560197	−	0.828359i	$-0.310725\pi$
$47$	7.68104	1.12039	0.560197	+	0.828359i	$0.310725\pi$
$48$	0	0
$49$	6.02683	0.860976
$50$	−1.66085	−0.234880
$51$	0	0
$52$	1.67097	0.231723
$53$	4.50651	0.619017	0.309509	−	0.950897i	$-0.399835\pi$
$53$	4.50651	0.619017	0.309509	+	0.950897i	$0.399835\pi$
$54$	0	0
$55$	0.456756	0.0615890
$56$	−7.04008	−0.940770
$57$	0	0
$58$	3.76068	0.493801
$59$	−2.41312	−0.314161	−0.157081	−	0.987586i	$-0.550208\pi$
$59$	−2.41312	−0.314161	−0.157081	+	0.987586i	$0.550208\pi$
$60$	0	0
$61$	6.37922	0.816775	0.408387	−	0.912809i	$-0.366091\pi$
$61$	6.37922	0.816775	0.408387	+	0.912809i	$0.366091\pi$
$62$	5.34441	0.678741
$63$	0	0
$64$	−2.15308	−0.269136
$65$	−1.30878	−0.162334
$66$	0	0
$67$	0.799497	0.0976741	0.0488370	−	0.998807i	$-0.484449\pi$
$67$	0.799497	0.0976741	0.0488370	+	0.998807i	$0.484449\pi$
$68$	2.15294	0.261082
$69$	0	0
$70$	2.55425	0.305291
$71$	−2.93852	−0.348738	−0.174369	−	0.984680i	$-0.555789\pi$
$71$	−2.93852	−0.348738	−0.174369	+	0.984680i	$0.555789\pi$
$72$	0	0
$73$	−15.3091	−1.79180	−0.895899	−	0.444257i	$-0.853468\pi$
$73$	−15.3091	−1.79180	−0.895899	+	0.444257i	$0.853468\pi$
$74$	2.85574	0.331973
$75$	0	0
$76$	9.49353	1.08898
$77$	1.21950	0.138975
$78$	0	0
$79$	10.3913	1.16911	0.584557	−	0.811353i	$-0.301268\pi$
$79$	10.3913	1.16911	0.584557	+	0.811353i	$0.301268\pi$
$80$	3.28597	0.367382
$81$	0	0
$82$	3.79321	0.418890
$83$	13.3213	1.46220	0.731102	−	0.682269i	$-0.239007\pi$
$83$	13.3213	1.46220	0.731102	+	0.682269i	$0.239007\pi$
$84$	0	0
$85$	−1.68627	−0.182902
$86$	−3.39715	−0.366324
$87$	0	0
$88$	−0.659054	−0.0702554
$89$	4.42608	0.469163	0.234582	−	0.972096i	$-0.424628\pi$
$89$	4.42608	0.469163	0.234582	+	0.972096i	$0.424628\pi$
$90$	0	0
$91$	−3.49433	−0.366305
$92$	5.41201	0.564241
$93$	0	0
$94$	4.02107	0.414742
$95$	−7.43573	−0.762890
$96$	0	0
$97$	−6.98217	−0.708932	−0.354466	−	0.935069i	$-0.615337\pi$
$97$	−6.98217	−0.708932	−0.354466	+	0.935069i	$0.615337\pi$
$98$	3.15508	0.318712
$99$	0	0
$100$	5.47565	0.547565
$101$	3.92586	0.390638	0.195319	−	0.980740i	$-0.437426\pi$
$101$	3.92586	0.390638	0.195319	+	0.980740i	$0.437426\pi$
$102$	0	0
$103$	−14.0937	−1.38870	−0.694348	−	0.719639i	$-0.744307\pi$
$103$	−14.0937	−1.38870	−0.694348	+	0.719639i	$0.744307\pi$
$104$	1.88843	0.185176
$105$	0	0
$106$	2.35919	0.229145
$107$	−7.65150	−0.739699	−0.369849	−	0.929092i	$-0.620591\pi$
$107$	−7.65150	−0.739699	−0.369849	+	0.929092i	$0.620591\pi$
$108$	0	0
$109$	−4.57688	−0.438386	−0.219193	−	0.975682i	$-0.570342\pi$
$109$	−4.57688	−0.438386	−0.219193	+	0.975682i	$0.570342\pi$
$110$	0.239115	0.0227987
$111$	0	0
$112$	8.77325	0.828994
$113$	15.9981	1.50497	0.752487	−	0.658607i	$-0.228854\pi$
$113$	15.9981	1.50497	0.752487	+	0.658607i	$0.228854\pi$
$114$	0	0
$115$	−4.23891	−0.395281
$116$	−12.3985	−1.15118
$117$	0	0
$118$	−1.26328	−0.116295
$119$	−4.50220	−0.412716
$120$	0	0
$121$	−10.8858	−0.989622
$122$	3.33956	0.302350
$123$	0	0
$124$	−17.6199	−1.58231
$125$	−11.0479	−0.988154
$126$	0	0
$127$	7.02400	0.623279	0.311640	−	0.950200i	$-0.399122\pi$
$127$	7.02400	0.623279	0.311640	+	0.950200i	$0.399122\pi$
$128$	−11.4744	−1.01420
$129$	0	0
$130$	−0.685153	−0.0600919
$131$	3.36814	0.294276	0.147138	−	0.989116i	$-0.452994\pi$
$131$	3.36814	0.294276	0.147138	+	0.989116i	$0.452994\pi$
$132$	0	0
$133$	−19.8528	−1.72145
$134$	0.418542	0.0361565
$135$	0	0
$136$	2.43312	0.208638
$137$	21.8903	1.87022	0.935108	−	0.354362i	$-0.115302\pi$
$137$	21.8903	1.87022	0.935108	+	0.354362i	$0.115302\pi$
$138$	0	0
$139$	−7.81159	−0.662571	−0.331285	−	0.943531i	$-0.607482\pi$
$139$	−7.81159	−0.662571	−0.331285	+	0.943531i	$0.607482\pi$
$140$	−8.42107	−0.711710
$141$	0	0
$142$	−1.53833	−0.129094
$143$	−0.327120	−0.0273551
$144$	0	0
$145$	9.71105	0.806459
$146$	−8.01443	−0.663279
$147$	0	0
$148$	−9.41504	−0.773911
$149$	1.73671	0.142277	0.0711385	−	0.997466i	$-0.477337\pi$
$149$	1.73671	0.142277	0.0711385	+	0.997466i	$0.477337\pi$
$150$	0	0
$151$	9.10500	0.740955	0.370477	−	0.928841i	$-0.379194\pi$
$151$	9.10500	0.740955	0.370477	+	0.928841i	$0.379194\pi$
$152$	10.7290	0.870237
$153$	0	0
$154$	0.638417	0.0514451
$155$	13.8007	1.10850
$156$	0	0
$157$	10.4340	0.832727	0.416364	−	0.909198i	$-0.363304\pi$
$157$	10.4340	0.832727	0.416364	+	0.909198i	$0.363304\pi$
$158$	5.43992	0.432777
$159$	0	0
$160$	6.99386	0.552913
$161$	−11.3175	−0.891947
$162$	0	0
$163$	23.6947	1.85591	0.927954	−	0.372694i	$-0.121566\pi$
$163$	23.6947	1.85591	0.927954	+	0.372694i	$0.121566\pi$
$164$	−12.5058	−0.976537
$165$	0	0
$166$	6.97379	0.541271
$167$	14.5777	1.12806	0.564029	−	0.825755i	$-0.309251\pi$
$167$	14.5777	1.12806	0.564029	+	0.825755i	$0.309251\pi$
$168$	0	0
$169$	−12.0627	−0.927898
$170$	−0.882774	−0.0677057
$171$	0	0
$172$	11.2000	0.853993
$173$	16.5647	1.25939	0.629695	−	0.776843i	$-0.283180\pi$
$173$	16.5647	1.25939	0.629695	+	0.776843i	$0.283180\pi$
$174$	0	0
$175$	−11.4506	−0.865585
$176$	0.821304	0.0619081
$177$	0	0
$178$	2.31708	0.173672
$179$	−14.0810	−1.05246	−0.526230	−	0.850342i	$-0.676395\pi$
$179$	−14.0810	−1.05246	−0.526230	+	0.850342i	$0.676395\pi$
$180$	0	0
$181$	−5.40061	−0.401424	−0.200712	−	0.979650i	$-0.564326\pi$
$181$	−5.40061	−0.401424	−0.200712	+	0.979650i	$0.564326\pi$
$182$	−1.82930	−0.135597
$183$	0	0
$184$	6.11632	0.450901
$185$	7.37425	0.542166
$186$	0	0
$187$	−0.421472	−0.0308211
$188$	−13.2570	−0.966867
$189$	0	0
$190$	−3.89265	−0.282403
$191$	−7.86102	−0.568803	−0.284402	−	0.958705i	$-0.591795\pi$
$191$	−7.86102	−0.568803	−0.284402	+	0.958705i	$0.591795\pi$
$192$	0	0
$193$	23.5348	1.69407	0.847036	−	0.531536i	$-0.178385\pi$
$193$	23.5348	1.69407	0.847036	+	0.531536i	$0.178385\pi$
$194$	−3.65521	−0.262429
$195$	0	0
$196$	−10.4020	−0.742997
$197$	1.91517	0.136450	0.0682251	−	0.997670i	$-0.478266\pi$
$197$	1.91517	0.136450	0.0682251	+	0.997670i	$0.478266\pi$
$198$	0	0
$199$	2.24700	0.159286	0.0796430	−	0.996823i	$-0.474622\pi$
$199$	2.24700	0.159286	0.0796430	+	0.996823i	$0.474622\pi$
$200$	6.18824	0.437575
$201$	0	0
$202$	2.05521	0.144604
$203$	25.9277	1.81977
$204$	0	0
$205$	9.79505	0.684116
$206$	−7.37816	−0.514061
$207$	0	0
$208$	−2.35334	−0.163175
$209$	−1.85851	−0.128556
$210$	0	0
$211$	14.5078	0.998758	0.499379	−	0.866384i	$-0.333561\pi$
$211$	14.5078	0.998758	0.499379	+	0.866384i	$0.333561\pi$
$212$	−7.77797	−0.534193
$213$	0	0
$214$	−4.00561	−0.273818
$215$	−8.77232	−0.598267
$216$	0	0
$217$	36.8466	2.50131
$218$	−2.39603	−0.162279
$219$	0	0
$220$	−0.788335	−0.0531495
$221$	1.20767	0.0812370
$222$	0	0
$223$	−25.0935	−1.68038	−0.840191	−	0.542290i	$-0.817557\pi$
$223$	−25.0935	−1.68038	−0.840191	+	0.542290i	$0.817557\pi$
$224$	18.6730	1.24764
$225$	0	0
$226$	8.37511	0.557104
$227$	20.1566	1.33784	0.668922	−	0.743333i	$-0.266756\pi$
$227$	20.1566	1.33784	0.668922	+	0.743333i	$0.266756\pi$
$228$	0	0
$229$	24.6507	1.62897	0.814483	−	0.580188i	$-0.197021\pi$
$229$	24.6507	1.62897	0.814483	+	0.580188i	$0.197021\pi$
$230$	−2.21910	−0.146323
$231$	0	0
$232$	−14.0121	−0.919938
$233$	22.2305	1.45637	0.728185	−	0.685380i	$-0.240364\pi$
$233$	22.2305	1.45637	0.728185	+	0.685380i	$0.240364\pi$
$234$	0	0
$235$	10.3834	0.677342
$236$	4.16490	0.271112
$237$	0	0
$238$	−2.35693	−0.152777
$239$	5.88759	0.380837	0.190418	−	0.981703i	$-0.439016\pi$
$239$	5.88759	0.380837	0.190418	+	0.981703i	$0.439016\pi$
$240$	0	0
$241$	7.97674	0.513827	0.256913	−	0.966434i	$-0.417294\pi$
$241$	7.97674	0.513827	0.256913	+	0.966434i	$0.417294\pi$
$242$	−5.69881	−0.366333
$243$	0	0
$244$	−11.0102	−0.704853
$245$	8.14725	0.520508
$246$	0	0
$247$	5.32532	0.338842
$248$	−19.9130	−1.26447
$249$	0	0
$250$	−5.78365	−0.365790
$251$	20.2547	1.27847	0.639233	−	0.769013i	$-0.279252\pi$
$251$	20.2547	1.27847	0.639233	+	0.769013i	$0.279252\pi$
$252$	0	0
$253$	−1.05949	−0.0666093
$254$	3.67711	0.230722
$255$	0	0
$256$	−1.70075	−0.106297
$257$	15.9624	0.995705	0.497853	−	0.867262i	$-0.334122\pi$
$257$	15.9624	0.995705	0.497853	+	0.867262i	$0.334122\pi$
$258$	0	0
$259$	19.6886	1.22339
$260$	2.25887	0.140089
$261$	0	0
$262$	1.76324	0.108933
$263$	−26.2647	−1.61955	−0.809774	−	0.586741i	$-0.800411\pi$
$263$	−26.2647	−1.61955	−0.809774	+	0.586741i	$0.800411\pi$
$264$	0	0
$265$	6.09204	0.374231
$266$	−10.3930	−0.637239
$267$	0	0
$268$	−1.37988	−0.0842898
$269$	−7.56302	−0.461125	−0.230563	−	0.973057i	$-0.574057\pi$
$269$	−7.56302	−0.461125	−0.230563	+	0.973057i	$0.574057\pi$
$270$	0	0
$271$	0.855122	0.0519450	0.0259725	−	0.999663i	$-0.491732\pi$
$271$	0.855122	0.0519450	0.0259725	+	0.999663i	$0.491732\pi$
$272$	−3.03212	−0.183850
$273$	0	0
$274$	11.4597	0.692307
$275$	−1.07194	−0.0646407
$276$	0	0
$277$	−7.71774	−0.463714	−0.231857	−	0.972750i	$-0.574480\pi$
$277$	−7.71774	−0.463714	−0.231857	+	0.972750i	$0.574480\pi$
$278$	−4.08942	−0.245267
$279$	0	0
$280$	−9.51698	−0.568748
$281$	−5.04416	−0.300910	−0.150455	−	0.988617i	$-0.548074\pi$
$281$	−5.04416	−0.300910	−0.150455	+	0.988617i	$0.548074\pi$
$282$	0	0
$283$	1.44488	0.0858894	0.0429447	−	0.999077i	$-0.486326\pi$
$283$	1.44488	0.0858894	0.0429447	+	0.999077i	$0.486326\pi$
$284$	5.07171	0.300950
$285$	0	0
$286$	−0.171249	−0.0101262
$287$	26.1519	1.54370
$288$	0	0
$289$	−15.4440	−0.908470
$290$	5.08380	0.298531
$291$	0	0
$292$	26.4227	1.54627
$293$	−11.9975	−0.700902	−0.350451	−	0.936581i	$-0.613972\pi$
$293$	−11.9975	−0.700902	−0.350451	+	0.936581i	$0.613972\pi$
$294$	0	0
$295$	−3.26213	−0.189928
$296$	−10.6403	−0.618455
$297$	0	0
$298$	0.909180	0.0526674
$299$	3.03582	0.175566
$300$	0	0
$301$	−23.4213	−1.34998
$302$	4.76653	0.274283
$303$	0	0
$304$	−13.3703	−0.766842
$305$	8.62361	0.493787
$306$	0	0
$307$	14.7380	0.841140	0.420570	−	0.907260i	$-0.361830\pi$
$307$	14.7380	0.841140	0.420570	+	0.907260i	$0.361830\pi$
$308$	−2.10479	−0.119931
$309$	0	0
$310$	7.22473	0.410337
$311$	25.6282	1.45324	0.726621	−	0.687038i	$-0.241090\pi$
$311$	25.6282	1.45324	0.726621	+	0.687038i	$0.241090\pi$
$312$	0	0
$313$	−3.29616	−0.186310	−0.0931551	−	0.995652i	$-0.529695\pi$
$313$	−3.29616	−0.186310	−0.0931551	+	0.995652i	$0.529695\pi$
$314$	5.46229	0.308255
$315$	0	0
$316$	−17.9348	−1.00891
$317$	−21.5851	−1.21234	−0.606170	−	0.795335i	$-0.707295\pi$
$317$	−21.5851	−1.21234	−0.606170	+	0.795335i	$0.707295\pi$
$318$	0	0
$319$	2.42721	0.135898
$320$	−2.91060	−0.162708
$321$	0	0
$322$	−5.92480	−0.330176
$323$	6.86132	0.381774
$324$	0	0
$325$	3.07152	0.170377
$326$	12.4043	0.687011
$327$	0	0
$328$	−14.1333	−0.780379
$329$	27.7229	1.52841
$330$	0	0
$331$	−19.2871	−1.06011	−0.530057	−	0.847962i	$-0.677829\pi$
$331$	−19.2871	−1.06011	−0.530057	+	0.847962i	$0.677829\pi$
$332$	−22.9918	−1.26184
$333$	0	0
$334$	7.63153	0.417579
$335$	1.08078	0.0590495
$336$	0	0
$337$	0.183297	0.00998481	0.00499241	−	0.999988i	$-0.498411\pi$
$337$	0.183297	0.00998481	0.00499241	+	0.999988i	$0.498411\pi$
$338$	−6.31489	−0.343485
$339$	0	0
$340$	2.91041	0.157839
$341$	3.44938	0.186794
$342$	0	0
$343$	−3.51243	−0.189654
$344$	12.6576	0.682451
$345$	0	0
$346$	8.67172	0.466194
$347$	24.5220	1.31641	0.658206	−	0.752838i	$-0.271316\pi$
$347$	24.5220	1.31641	0.658206	+	0.752838i	$0.271316\pi$
$348$	0	0
$349$	−8.19072	−0.438439	−0.219220	−	0.975676i	$-0.570351\pi$
$349$	−8.19072	−0.438439	−0.219220	+	0.975676i	$0.570351\pi$
$350$	−5.99447	−0.320418
$351$	0	0
$352$	1.74807	0.0931722
$353$	−17.5573	−0.934483	−0.467242	−	0.884130i	$-0.654752\pi$
$353$	−17.5573	−0.934483	−0.467242	+	0.884130i	$0.654752\pi$
$354$	0	0
$355$	−3.97237	−0.210832
$356$	−7.63915	−0.404874
$357$	0	0
$358$	−7.37147	−0.389595
$359$	−20.7724	−1.09632	−0.548162	−	0.836372i	$-0.684672\pi$
$359$	−20.7724	−1.09632	−0.548162	+	0.836372i	$0.684672\pi$
$360$	0	0
$361$	11.2554	0.592391
$362$	−2.82726	−0.148597
$363$	0	0
$364$	6.03100	0.316110
$365$	−20.6953	−1.08324
$366$	0	0
$367$	−34.8797	−1.82070	−0.910352	−	0.413835i	$-0.864189\pi$
$367$	−34.8797	−1.82070	−0.910352	+	0.413835i	$0.864189\pi$
$368$	−7.62208	−0.397329
$369$	0	0
$370$	3.86047	0.200696
$371$	16.2652	0.844448
$372$	0	0
$373$	−9.73592	−0.504107	−0.252054	−	0.967713i	$-0.581106\pi$
$373$	−9.73592	−0.504107	−0.252054	+	0.967713i	$0.581106\pi$
$374$	−0.220643	−0.0114092
$375$	0	0
$376$	−14.9823	−0.772652
$377$	−6.95486	−0.358193
$378$	0	0
$379$	17.7671	0.912634	0.456317	−	0.889817i	$-0.349168\pi$
$379$	17.7671	0.912634	0.456317	+	0.889817i	$0.349168\pi$
$380$	12.8336	0.658351
$381$	0	0
$382$	−4.11529	−0.210557
$383$	−12.7845	−0.653257	−0.326628	−	0.945153i	$-0.605913\pi$
$383$	−12.7845	−0.653257	−0.326628	+	0.945153i	$0.605913\pi$
$384$	0	0
$385$	1.64856	0.0840183
$386$	12.3206	0.627103
$387$	0	0
$388$	12.0508	0.611787
$389$	10.8081	0.547992	0.273996	−	0.961731i	$-0.411655\pi$
$389$	10.8081	0.547992	0.273996	+	0.961731i	$0.411655\pi$
$390$	0	0
$391$	3.91146	0.197811
$392$	−11.7557	−0.593750
$393$	0	0
$394$	1.00260	0.0505105
$395$	14.0473	0.706796
$396$	0	0
$397$	−2.02404	−0.101584	−0.0507919	−	0.998709i	$-0.516175\pi$
$397$	−2.02404	−0.101584	−0.0507919	+	0.998709i	$0.516175\pi$
$398$	1.17632	0.0589636
$399$	0	0
$400$	−7.71171	−0.385585
$401$	7.98951	0.398977	0.199488	−	0.979900i	$-0.436072\pi$
$401$	7.98951	0.398977	0.199488	+	0.979900i	$0.436072\pi$
$402$	0	0
$403$	−9.88375	−0.492345
$404$	−6.77581	−0.337109
$405$	0	0
$406$	13.5733	0.673632
$407$	1.84314	0.0913612
$408$	0	0
$409$	−17.7234	−0.876366	−0.438183	−	0.898886i	$-0.644378\pi$
$409$	−17.7234	−0.876366	−0.438183	+	0.898886i	$0.644378\pi$
$410$	5.12777	0.253243
$411$	0	0
$412$	24.3250	1.19840
$413$	−8.70960	−0.428571
$414$	0	0
$415$	18.0081	0.883984
$416$	−5.00886	−0.245579
$417$	0	0
$418$	−0.972941	−0.0475881
$419$	−23.2376	−1.13523	−0.567615	−	0.823294i	$-0.692134\pi$
$419$	−23.2376	−1.13523	−0.567615	+	0.823294i	$0.692134\pi$
$420$	0	0
$421$	16.0445	0.781961	0.390980	−	0.920399i	$-0.372136\pi$
$421$	16.0445	0.781961	0.390980	+	0.920399i	$0.372136\pi$
$422$	7.59493	0.369715
$423$	0	0
$424$	−8.79020	−0.426890
$425$	3.95745	0.191964
$426$	0	0
$427$	23.0243	1.11422
$428$	13.2060	0.638338
$429$	0	0
$430$	−4.59236	−0.221463
$431$	−16.8476	−0.811519	−0.405759	−	0.913980i	$-0.632993\pi$
$431$	−16.8476	−0.811519	−0.405759	+	0.913980i	$0.632993\pi$
$432$	0	0
$433$	13.9110	0.668518	0.334259	−	0.942481i	$-0.391514\pi$
$433$	13.9110	0.668518	0.334259	+	0.942481i	$0.391514\pi$
$434$	19.2894	0.925922
$435$	0	0
$436$	7.89942	0.378314
$437$	17.2478	0.825075
$438$	0	0
$439$	36.1726	1.72643	0.863213	−	0.504840i	$-0.168448\pi$
$439$	36.1726	1.72643	0.863213	+	0.504840i	$0.168448\pi$
$440$	−0.890928	−0.0424733
$441$	0	0
$442$	0.632225	0.0300719
$443$	33.8300	1.60731	0.803655	−	0.595096i	$-0.202886\pi$
$443$	33.8300	1.60731	0.803655	+	0.595096i	$0.202886\pi$
$444$	0	0
$445$	5.98330	0.283636
$446$	−13.1366	−0.622036
$447$	0	0
$448$	−7.77106	−0.367148
$449$	35.4973	1.67522	0.837611	−	0.546267i	$-0.183951\pi$
$449$	35.4973	1.67522	0.837611	+	0.546267i	$0.183951\pi$
$450$	0	0
$451$	2.44820	0.115281
$452$	−27.6118	−1.29875
$453$	0	0
$454$	10.5521	0.495236
$455$	−4.72373	−0.221452
$456$	0	0
$457$	−27.6267	−1.29232	−0.646162	−	0.763201i	$-0.723627\pi$
$457$	−27.6267	−1.29232	−0.646162	+	0.763201i	$0.723627\pi$
$458$	12.9048	0.603002
$459$	0	0
$460$	7.31611	0.341116
$461$	1.39808	0.0651150	0.0325575	−	0.999470i	$-0.489635\pi$
$461$	1.39808	0.0651150	0.0325575	+	0.999470i	$0.489635\pi$
$462$	0	0
$463$	−16.4657	−0.765225	−0.382613	−	0.923909i	$-0.624976\pi$
$463$	−16.4657	−0.765225	−0.382613	+	0.923909i	$0.624976\pi$
$464$	17.4617	0.810637
$465$	0	0
$466$	11.6378	0.539112
$467$	−19.2949	−0.892863	−0.446432	−	0.894818i	$-0.647305\pi$
$467$	−19.2949	−0.892863	−0.446432	+	0.894818i	$0.647305\pi$
$468$	0	0
$469$	2.88560	0.133245
$470$	5.43580	0.250735
$471$	0	0
$472$	4.70692	0.216654
$473$	−2.19258	−0.100815
$474$	0	0
$475$	17.4506	0.800690
$476$	7.77054	0.356162
$477$	0	0
$478$	3.08219	0.140976
$479$	26.2032	1.19725	0.598626	−	0.801028i	$-0.295713\pi$
$479$	26.2032	1.19725	0.598626	+	0.801028i	$0.295713\pi$
$480$	0	0
$481$	−5.28129	−0.240806
$482$	4.17587	0.190206
$483$	0	0
$484$	18.7883	0.854014
$485$	−9.43870	−0.428589
$486$	0	0
$487$	30.4664	1.38057	0.690283	−	0.723539i	$-0.257486\pi$
$487$	30.4664	1.38057	0.690283	+	0.723539i	$0.257486\pi$
$488$	−12.4430	−0.563268
$489$	0	0
$490$	4.26514	0.192679
$491$	−36.3956	−1.64251	−0.821255	−	0.570562i	$-0.806726\pi$
$491$	−36.3956	−1.64251	−0.821255	+	0.570562i	$0.806726\pi$
$492$	0	0
$493$	−8.96087	−0.403577
$494$	2.78784	0.125431
$495$	0	0
$496$	24.8153	1.11424
$497$	−10.6059	−0.475740
$498$	0	0
$499$	−2.03599	−0.0911435	−0.0455717	−	0.998961i	$-0.514511\pi$
$499$	−2.03599	−0.0911435	−0.0455717	+	0.998961i	$0.514511\pi$
$500$	19.0680	0.852748
$501$	0	0
$502$	10.6035	0.473256
$503$	−23.3684	−1.04195	−0.520973	−	0.853573i	$-0.674431\pi$
$503$	−23.3684	−1.04195	−0.520973	+	0.853573i	$0.674431\pi$
$504$	0	0
$505$	5.30709	0.236163
$506$	−0.554648	−0.0246571
$507$	0	0
$508$	−12.1230	−0.537872
$509$	−32.6744	−1.44827	−0.724133	−	0.689661i	$-0.757760\pi$
$509$	−32.6744	−1.44827	−0.724133	+	0.689661i	$0.757760\pi$
$510$	0	0
$511$	−55.2548	−2.44433
$512$	22.0584	0.974855
$513$	0	0
$514$	8.35640	0.368585
$515$	−19.0523	−0.839546
$516$	0	0
$517$	2.59527	0.114140
$518$	10.3071	0.452869
$519$	0	0
$520$	2.55284	0.111949
$521$	−8.73632	−0.382745	−0.191373	−	0.981517i	$-0.561294\pi$
$521$	−8.73632	−0.382745	−0.191373	+	0.981517i	$0.561294\pi$
$522$	0	0
$523$	15.6850	0.685855	0.342928	−	0.939362i	$-0.388581\pi$
$523$	15.6850	0.685855	0.342928	+	0.939362i	$0.388581\pi$
$524$	−5.81321	−0.253951
$525$	0	0
$526$	−13.7497	−0.599516
$527$	−12.7346	−0.554726
$528$	0	0
$529$	−13.1675	−0.572499
$530$	3.18922	0.138531
$531$	0	0
$532$	34.2647	1.48556
$533$	−7.01502	−0.303854
$534$	0	0
$535$	−10.3435	−0.447190
$536$	−1.55946	−0.0673585
$537$	0	0
$538$	−3.95929	−0.170697
$539$	2.03635	0.0877117
$540$	0	0
$541$	28.4519	1.22324	0.611621	−	0.791151i	$-0.290518\pi$
$541$	28.4519	1.22324	0.611621	+	0.791151i	$0.290518\pi$
$542$	0.447662	0.0192287
$543$	0	0
$544$	−6.45358	−0.276695
$545$	−6.18716	−0.265029
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−37.7814	−1.61394
$549$	0	0
$550$	−0.561170	−0.0239284
$551$	−39.5135	−1.68333
$552$	0	0
$553$	37.5051	1.59488
$554$	−4.04029	−0.171655
$555$	0	0
$556$	13.4824	0.571779
$557$	−32.4488	−1.37490	−0.687450	−	0.726231i	$-0.741270\pi$
$557$	−32.4488	−1.37490	−0.687450	+	0.726231i	$0.741270\pi$
$558$	0	0
$559$	6.28256	0.265724
$560$	11.8599	0.501174
$561$	0	0
$562$	−2.64065	−0.111389
$563$	−23.7803	−1.00222	−0.501111	−	0.865383i	$-0.667075\pi$
$563$	−23.7803	−1.00222	−0.501111	+	0.865383i	$0.667075\pi$
$564$	0	0
$565$	21.6267	0.909842
$566$	0.756406	0.0317941
$567$	0	0
$568$	5.73173	0.240498
$569$	−42.0011	−1.76078	−0.880389	−	0.474252i	$-0.842719\pi$
$569$	−42.0011	−1.76078	−0.880389	+	0.474252i	$0.842719\pi$
$570$	0	0
$571$	−39.1170	−1.63699	−0.818497	−	0.574510i	$-0.805193\pi$
$571$	−39.1170	−1.63699	−0.818497	+	0.574510i	$0.805193\pi$
$572$	0.564589	0.0236067
$573$	0	0
$574$	13.6907	0.571439
$575$	9.94814	0.414866
$576$	0	0
$577$	−20.9523	−0.872256	−0.436128	−	0.899885i	$-0.643651\pi$
$577$	−20.9523	−0.872256	−0.436128	+	0.899885i	$0.643651\pi$
$578$	−8.08503	−0.336293
$579$	0	0
$580$	−16.7607	−0.695950
$581$	48.0802	1.99470
$582$	0	0
$583$	1.52266	0.0630622
$584$	29.8613	1.23567
$585$	0	0
$586$	−6.28077	−0.259456
$587$	37.1311	1.53257	0.766283	−	0.642503i	$-0.222104\pi$
$587$	37.1311	1.53257	0.766283	+	0.642503i	$0.222104\pi$
$588$	0	0
$589$	−56.1538	−2.31378
$590$	−1.70774	−0.0703067
$591$	0	0
$592$	13.2598	0.544975
$593$	−19.1878	−0.787949	−0.393974	−	0.919121i	$-0.628900\pi$
$593$	−19.1878	−0.787949	−0.393974	+	0.919121i	$0.628900\pi$
$594$	0	0
$595$	−6.08621	−0.249510
$596$	−2.99746	−0.122781
$597$	0	0
$598$	1.58927	0.0649902
$599$	−31.5960	−1.29098	−0.645488	−	0.763770i	$-0.723346\pi$
$599$	−31.5960	−1.29098	−0.645488	+	0.763770i	$0.723346\pi$
$600$	0	0
$601$	16.9578	0.691725	0.345862	−	0.938285i	$-0.387586\pi$
$601$	16.9578	0.691725	0.345862	+	0.938285i	$0.387586\pi$
$602$	−12.2612	−0.499730
$603$	0	0
$604$	−15.7147	−0.639422
$605$	−14.7158	−0.598282
$606$	0	0
$607$	−10.5096	−0.426572	−0.213286	−	0.976990i	$-0.568417\pi$
$607$	−10.5096	−0.426572	−0.213286	+	0.976990i	$0.568417\pi$
$608$	−28.4575	−1.15410
$609$	0	0
$610$	4.51452	0.182787
$611$	−7.43642	−0.300845
$612$	0	0
$613$	−14.8071	−0.598055	−0.299027	−	0.954245i	$-0.596662\pi$
$613$	−14.8071	−0.598055	−0.299027	+	0.954245i	$0.596662\pi$
$614$	7.71542	0.311369
$615$	0	0
$616$	−2.37870	−0.0958407
$617$	27.3637	1.10162	0.550811	−	0.834630i	$-0.314319\pi$
$617$	27.3637	1.10162	0.550811	+	0.834630i	$0.314319\pi$
$618$	0	0
$619$	4.81738	0.193627	0.0968134	−	0.995303i	$-0.469135\pi$
$619$	4.81738	0.193627	0.0968134	+	0.995303i	$0.469135\pi$
$620$	−23.8191	−0.956599
$621$	0	0
$622$	13.4165	0.537954
$623$	15.9749	0.640021
$624$	0	0
$625$	0.927908	0.0371163
$626$	−1.72556	−0.0689674
$627$	0	0
$628$	−18.0085	−0.718619
$629$	−6.80459	−0.271317
$630$	0	0
$631$	29.9291	1.19146	0.595730	−	0.803185i	$-0.296863\pi$
$631$	29.9291	1.19146	0.595730	+	0.803185i	$0.296863\pi$
$632$	−20.2688	−0.806250
$633$	0	0
$634$	−11.2999	−0.448778
$635$	9.49525	0.376808
$636$	0	0
$637$	−5.83489	−0.231187
$638$	1.27066	0.0503059
$639$	0	0
$640$	−15.5114	−0.613143
$641$	−2.88534	−0.113964	−0.0569821	−	0.998375i	$-0.518148\pi$
$641$	−2.88534	−0.113964	−0.0569821	+	0.998375i	$0.518148\pi$
$642$	0	0
$643$	−32.7470	−1.29142	−0.645708	−	0.763584i	$-0.723438\pi$
$643$	−32.7470	−1.29142	−0.645708	+	0.763584i	$0.723438\pi$
$644$	19.5334	0.769724
$645$	0	0
$646$	3.59194	0.141323
$647$	−20.0383	−0.787787	−0.393894	−	0.919156i	$-0.628872\pi$
$647$	−20.0383	−0.787787	−0.393894	+	0.919156i	$0.628872\pi$
$648$	0	0
$649$	−0.815346	−0.0320051
$650$	1.60796	0.0630694
$651$	0	0
$652$	−40.8956	−1.60159
$653$	8.58315	0.335885	0.167942	−	0.985797i	$-0.446288\pi$
$653$	8.58315	0.335885	0.167942	+	0.985797i	$0.446288\pi$
$654$	0	0
$655$	4.55315	0.177906
$656$	17.6127	0.687660
$657$	0	0
$658$	14.5131	0.565781
$659$	34.5534	1.34601	0.673004	−	0.739638i	$-0.265003\pi$
$659$	34.5534	1.34601	0.673004	+	0.739638i	$0.265003\pi$
$660$	0	0
$661$	−18.6496	−0.725386	−0.362693	−	0.931909i	$-0.618143\pi$
$661$	−18.6496	−0.725386	−0.362693	+	0.931909i	$0.618143\pi$
$662$	−10.0969	−0.392428
$663$	0	0
$664$	−25.9839	−1.00837
$665$	−26.8375	−1.04072
$666$	0	0
$667$	−22.5256	−0.872196
$668$	−25.1603	−0.973480
$669$	0	0
$670$	0.565797	0.0218586
$671$	2.15541	0.0832087
$672$	0	0
$673$	23.0907	0.890083	0.445041	−	0.895510i	$-0.353189\pi$
$673$	23.0907	0.890083	0.445041	+	0.895510i	$0.353189\pi$
$674$	0.0959571	0.00369613
$675$	0	0
$676$	20.8195	0.800749
$677$	41.7475	1.60449	0.802243	−	0.596997i	$-0.203640\pi$
$677$	41.7475	1.60449	0.802243	+	0.596997i	$0.203640\pi$
$678$	0	0
$679$	−25.2005	−0.967108
$680$	3.28917	0.126134
$681$	0	0
$682$	1.80577	0.0691465
$683$	46.2392	1.76930	0.884648	−	0.466260i	$-0.154399\pi$
$683$	46.2392	1.76930	0.884648	+	0.466260i	$0.154399\pi$
$684$	0	0
$685$	29.5920	1.13065
$686$	−1.83878	−0.0702050
$687$	0	0
$688$	−15.7737	−0.601367
$689$	−4.36299	−0.166217
$690$	0	0
$691$	−51.8556	−1.97268	−0.986340	−	0.164724i	$-0.947327\pi$
$691$	−51.8556	−1.97268	−0.986340	+	0.164724i	$0.947327\pi$
$692$	−28.5897	−1.08682
$693$	0	0
$694$	12.8374	0.487303
$695$	−10.5599	−0.400561
$696$	0	0
$697$	−9.03838	−0.342353
$698$	−4.28790	−0.162299
$699$	0	0
$700$	19.7631	0.746975
$701$	−46.8098	−1.76798	−0.883992	−	0.467503i	$-0.845154\pi$
$701$	−46.8098	−1.76798	−0.883992	+	0.467503i	$0.845154\pi$
$702$	0	0
$703$	−30.0053	−1.13167
$704$	−0.727485	−0.0274181
$705$	0	0
$706$	−9.19138	−0.345922
$707$	14.1695	0.532899
$708$	0	0
$709$	2.30760	0.0866637	0.0433318	−	0.999061i	$-0.486203\pi$
$709$	2.30760	0.0866637	0.0433318	+	0.999061i	$0.486203\pi$
$710$	−2.07956	−0.0780446
$711$	0	0
$712$	−8.63330	−0.323547
$713$	−32.0118	−1.19885
$714$	0	0
$715$	−0.442210	−0.0165377
$716$	24.3029	0.908242
$717$	0	0
$718$	−10.8745	−0.405831
$719$	−1.47404	−0.0549723	−0.0274862	−	0.999622i	$-0.508750\pi$
$719$	−1.47404	−0.0549723	−0.0274862	+	0.999622i	$0.508750\pi$
$720$	0	0
$721$	−50.8681	−1.89443
$722$	5.89229	0.219288
$723$	0	0
$724$	9.32114	0.346417
$725$	−22.7905	−0.846418
$726$	0	0
$727$	−11.7670	−0.436412	−0.218206	−	0.975903i	$-0.570021\pi$
$727$	−11.7670	−0.436412	−0.218206	+	0.975903i	$0.570021\pi$
$728$	6.81587	0.252613
$729$	0	0
$730$	−10.8341	−0.400990
$731$	8.09466	0.299392
$732$	0	0
$733$	20.5086	0.757502	0.378751	−	0.925499i	$-0.376354\pi$
$733$	20.5086	0.757502	0.378751	+	0.925499i	$0.376354\pi$
$734$	−18.2597	−0.673979
$735$	0	0
$736$	−16.2229	−0.597983
$737$	0.270134	0.00995052
$738$	0	0
$739$	−31.6389	−1.16385	−0.581927	−	0.813241i	$-0.697701\pi$
$739$	−31.6389	−1.16385	−0.581927	+	0.813241i	$0.697701\pi$
$740$	−12.7275	−0.467873
$741$	0	0
$742$	8.51495	0.312594
$743$	40.5825	1.48883	0.744414	−	0.667718i	$-0.232729\pi$
$743$	40.5825	1.48883	0.744414	+	0.667718i	$0.232729\pi$
$744$	0	0
$745$	2.34774	0.0860145
$746$	−5.09682	−0.186608
$747$	0	0
$748$	0.727436	0.0265977
$749$	−27.6163	−1.00908
$750$	0	0
$751$	27.4293	1.00091	0.500454	−	0.865763i	$-0.333166\pi$
$751$	27.4293	1.00091	0.500454	+	0.865763i	$0.333166\pi$
$752$	18.6707	0.680851
$753$	0	0
$754$	−3.64091	−0.132594
$755$	12.3084	0.447949
$756$	0	0
$757$	−19.8595	−0.721805	−0.360903	−	0.932603i	$-0.617531\pi$
$757$	−19.8595	−0.721805	−0.360903	+	0.932603i	$0.617531\pi$
$758$	9.30119	0.337834
$759$	0	0
$760$	14.5038	0.526108
$761$	30.0646	1.08984	0.544920	−	0.838488i	$-0.316560\pi$
$761$	30.0646	1.08984	0.544920	+	0.838488i	$0.316560\pi$
$762$	0	0
$763$	−16.5192	−0.598035
$764$	13.5677	0.490861
$765$	0	0
$766$	−6.69276	−0.241819
$767$	2.33627	0.0843578
$768$	0	0
$769$	−27.2458	−0.982510	−0.491255	−	0.871016i	$-0.663462\pi$
$769$	−27.2458	−0.982510	−0.491255	+	0.871016i	$0.663462\pi$
$770$	0.863030	0.0311015
$771$	0	0
$772$	−40.6197	−1.46193
$773$	42.4735	1.52767	0.763833	−	0.645414i	$-0.223315\pi$
$773$	42.4735	1.52767	0.763833	+	0.645414i	$0.223315\pi$
$774$	0	0
$775$	−32.3882	−1.16342
$776$	13.6191	0.488897
$777$	0	0
$778$	5.65810	0.202853
$779$	−39.8553	−1.42796
$780$	0	0
$781$	−0.992867	−0.0355276
$782$	2.04767	0.0732246
$783$	0	0
$784$	14.6497	0.523205
$785$	14.1050	0.503431
$786$	0	0
$787$	45.4020	1.61841	0.809203	−	0.587529i	$-0.199899\pi$
$787$	45.4020	1.61841	0.809203	+	0.587529i	$0.199899\pi$
$788$	−3.30547	−0.117753
$789$	0	0
$790$	7.35384	0.261638
$791$	57.7414	2.05305
$792$	0	0
$793$	−6.17606	−0.219318
$794$	−1.05960	−0.0376038
$795$	0	0
$796$	−3.87820	−0.137459
$797$	−27.6932	−0.980944	−0.490472	−	0.871457i	$-0.663176\pi$
$797$	−27.6932	−0.980944	−0.490472	+	0.871457i	$0.663176\pi$
$798$	0	0
$799$	−9.58133	−0.338963
$800$	−16.4136	−0.580309
$801$	0	0
$802$	4.18256	0.147691
$803$	−5.17265	−0.182539
$804$	0	0
$805$	−15.2994	−0.539232
$806$	−5.17421	−0.182254
$807$	0	0
$808$	−7.65760	−0.269394
$809$	−12.6422	−0.444475	−0.222238	−	0.974993i	$-0.571336\pi$
$809$	−12.6422	−0.444475	−0.222238	+	0.974993i	$0.571336\pi$
$810$	0	0
$811$	−7.30807	−0.256621	−0.128310	−	0.991734i	$-0.540955\pi$
$811$	−7.30807	−0.256621	−0.128310	+	0.991734i	$0.540955\pi$
$812$	−44.7497	−1.57041
$813$	0	0
$814$	0.964897	0.0338196
$815$	32.0311	1.12200
$816$	0	0
$817$	35.6939	1.24877
$818$	−9.27831	−0.324409
$819$	0	0
$820$	−16.9057	−0.590372
$821$	−36.6884	−1.28043	−0.640217	−	0.768194i	$-0.721156\pi$
$821$	−36.6884	−1.28043	−0.640217	+	0.768194i	$0.721156\pi$
$822$	0	0
$823$	−4.83046	−0.168379	−0.0841897	−	0.996450i	$-0.526830\pi$
$823$	−4.83046	−0.168379	−0.0841897	+	0.996450i	$0.526830\pi$
$824$	27.4906	0.957680
$825$	0	0
$826$	−4.55953	−0.158646
$827$	23.1753	0.805883	0.402942	−	0.915226i	$-0.367988\pi$
$827$	23.1753	0.805883	0.402942	+	0.915226i	$0.367988\pi$
$828$	0	0
$829$	27.0086	0.938046	0.469023	−	0.883186i	$-0.344606\pi$
$829$	27.0086	0.938046	0.469023	+	0.883186i	$0.344606\pi$
$830$	9.42737	0.327229
$831$	0	0
$832$	2.08451	0.0722676
$833$	−7.51787	−0.260479
$834$	0	0
$835$	19.7066	0.681974
$836$	3.20768	0.110940
$837$	0	0
$838$	−12.1650	−0.420234
$839$	19.5336	0.674373	0.337187	−	0.941438i	$-0.390525\pi$
$839$	19.5336	0.674373	0.337187	+	0.941438i	$0.390525\pi$
$840$	0	0
$841$	22.6046	0.779470
$842$	8.39939	0.289462
$843$	0	0
$844$	−25.0396	−0.861899
$845$	−16.3067	−0.560967
$846$	0	0
$847$	−39.2899	−1.35002
$848$	10.9542	0.376170
$849$	0	0
$850$	2.07175	0.0710604
$851$	−17.1052	−0.586359
$852$	0	0
$853$	38.7260	1.32595	0.662977	−	0.748640i	$-0.269293\pi$
$853$	38.7260	1.32595	0.662977	+	0.748640i	$0.269293\pi$
$854$	12.0534	0.412458
$855$	0	0
$856$	14.9247	0.510115
$857$	−35.3753	−1.20840	−0.604199	−	0.796834i	$-0.706507\pi$
$857$	−35.3753	−1.20840	−0.604199	+	0.796834i	$0.706507\pi$
$858$	0	0
$859$	7.81176	0.266534	0.133267	−	0.991080i	$-0.457453\pi$
$859$	7.81176	0.266534	0.133267	+	0.991080i	$0.457453\pi$
$860$	15.1405	0.516287
$861$	0	0
$862$	−8.81981	−0.300404
$863$	−7.24097	−0.246486	−0.123243	−	0.992377i	$-0.539329\pi$
$863$	−7.24097	−0.246486	−0.123243	+	0.992377i	$0.539329\pi$
$864$	0	0
$865$	22.3926	0.761372
$866$	7.28247	0.247468
$867$	0	0
$868$	−63.5950	−2.15856
$869$	3.51102	0.119103
$870$	0	0
$871$	−0.774035	−0.0262272
$872$	8.92745	0.302322
$873$	0	0
$874$	9.02934	0.305422
$875$	−39.8749	−1.34802
$876$	0	0
$877$	−27.0100	−0.912062	−0.456031	−	0.889964i	$-0.650729\pi$
$877$	−27.0100	−0.912062	−0.456031	+	0.889964i	$0.650729\pi$
$878$	18.9366	0.639080
$879$	0	0
$880$	1.11026	0.0374270
$881$	−20.0865	−0.676731	−0.338366	−	0.941015i	$-0.609874\pi$
$881$	−20.0865	−0.676731	−0.338366	+	0.941015i	$0.609874\pi$
$882$	0	0
$883$	11.1980	0.376842	0.188421	−	0.982088i	$-0.439663\pi$
$883$	11.1980	0.376842	0.188421	+	0.982088i	$0.439663\pi$
$884$	−2.08437	−0.0701051
$885$	0	0
$886$	17.7102	0.594986
$887$	−56.2189	−1.88765	−0.943824	−	0.330450i	$-0.892800\pi$
$887$	−56.2189	−1.88765	−0.943824	+	0.330450i	$0.892800\pi$
$888$	0	0
$889$	25.3515	0.850263
$890$	3.13230	0.104995
$891$	0	0
$892$	43.3098	1.45012
$893$	−42.2495	−1.41382
$894$	0	0
$895$	−19.0351	−0.636272
$896$	−41.4142	−1.38355
$897$	0	0
$898$	18.5831	0.620125
$899$	73.3368	2.44592
$900$	0	0
$901$	−5.62143	−0.187277
$902$	1.28165	0.0426743
$903$	0	0
$904$	−31.2051	−1.03787
$905$	−7.30071	−0.242684
$906$	0	0
$907$	−41.1969	−1.36792	−0.683960	−	0.729520i	$-0.739744\pi$
$907$	−41.1969	−1.36792	−0.683960	+	0.729520i	$0.739744\pi$
$908$	−34.7892	−1.15452
$909$	0	0
$910$	−2.47290	−0.0819760
$911$	−26.4943	−0.877795	−0.438897	−	0.898537i	$-0.644631\pi$
$911$	−26.4943	−0.877795	−0.438897	+	0.898537i	$0.644631\pi$
$912$	0	0
$913$	4.50101	0.148962
$914$	−14.4628	−0.478386
$915$	0	0
$916$	−42.5457	−1.40575
$917$	12.1565	0.401444
$918$	0	0
$919$	−29.7090	−0.980008	−0.490004	−	0.871720i	$-0.663005\pi$
$919$	−29.7090	−0.980008	−0.490004	+	0.871720i	$0.663005\pi$
$920$	8.26823	0.272595
$921$	0	0
$922$	0.731902	0.0241039
$923$	2.84493	0.0936421
$924$	0	0
$925$	−17.3064	−0.569029
$926$	−8.61989	−0.283267
$927$	0	0
$928$	37.1654	1.22002
$929$	−31.6127	−1.03718	−0.518589	−	0.855024i	$-0.673543\pi$
$929$	−31.6127	−1.03718	−0.518589	+	0.855024i	$0.673543\pi$
$930$	0	0
$931$	−33.1505	−1.08646
$932$	−38.3686	−1.25680
$933$	0	0
$934$	−10.1010	−0.330516
$935$	−0.569758	−0.0186331
$936$	0	0
$937$	25.5752	0.835506	0.417753	−	0.908561i	$-0.362818\pi$
$937$	25.5752	0.835506	0.417753	+	0.908561i	$0.362818\pi$
$938$	1.51063	0.0493238
$939$	0	0
$940$	−17.9212	−0.584526
$941$	38.4356	1.25296	0.626482	−	0.779436i	$-0.284494\pi$
$941$	38.4356	1.25296	0.626482	+	0.779436i	$0.284494\pi$
$942$	0	0
$943$	−22.7205	−0.739880
$944$	−5.86570	−0.190912
$945$	0	0
$946$	−1.14783	−0.0373191
$947$	−39.7014	−1.29012	−0.645060	−	0.764132i	$-0.723168\pi$
$947$	−39.7014	−1.29012	−0.645060	+	0.764132i	$0.723168\pi$
$948$	0	0
$949$	14.8216	0.481129
$950$	9.13552	0.296395
$951$	0	0
$952$	8.78179	0.284620
$953$	22.9371	0.743007	0.371504	−	0.928432i	$-0.378842\pi$
$953$	22.9371	0.743007	0.371504	+	0.928432i	$0.378842\pi$
$954$	0	0
$955$	−10.6268	−0.343874
$956$	−10.1616	−0.328651
$957$	0	0
$958$	13.7175	0.443193
$959$	79.0081	2.55130
$960$	0	0
$961$	73.2211	2.36197
$962$	−2.76479	−0.0891404
$963$	0	0
$964$	−13.7674	−0.443417
$965$	31.8150	1.02416
$966$	0	0
$967$	−26.6775	−0.857891	−0.428946	−	0.903330i	$-0.641115\pi$
$967$	−26.6775	−0.857891	−0.428946	+	0.903330i	$0.641115\pi$
$968$	21.2334	0.682468
$969$	0	0
$970$	−4.94122	−0.158653
$971$	9.36106	0.300411	0.150205	−	0.988655i	$-0.452007\pi$
$971$	9.36106	0.300411	0.150205	+	0.988655i	$0.452007\pi$
$972$	0	0
$973$	−28.1942	−0.903863
$974$	15.9494	0.511051
$975$	0	0
$976$	15.5063	0.496345
$977$	−17.6428	−0.564445	−0.282222	−	0.959349i	$-0.591072\pi$
$977$	−17.6428	−0.564445	−0.282222	+	0.959349i	$0.591072\pi$
$978$	0	0
$979$	1.49548	0.0477959
$980$	−14.0617	−0.449183
$981$	0	0
$982$	−19.0533	−0.608016
$983$	1.76510	0.0562979	0.0281490	−	0.999604i	$-0.491039\pi$
$983$	1.76510	0.0562979	0.0281490	+	0.999604i	$0.491039\pi$
$984$	0	0
$985$	2.58898	0.0824919
$986$	−4.69107	−0.149394
$987$	0	0
$988$	−9.19119	−0.292411
$989$	20.3482	0.647034
$990$	0	0
$991$	−33.2753	−1.05702	−0.528512	−	0.848926i	$-0.677250\pi$
$991$	−33.2753	−1.05702	−0.528512	+	0.848926i	$0.677250\pi$
$992$	52.8169	1.67694
$993$	0	0
$994$	−5.55226	−0.176107
$995$	3.03757	0.0962973
$996$	0	0
$997$	10.1732	0.322187	0.161094	−	0.986939i	$-0.448498\pi$
$997$	10.1732	0.322187	0.161094	+	0.986939i	$0.448498\pi$
$998$	−1.06585	−0.0337390
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.9		18
3.2	odd	2		547.2.a.b.1.10	✓	18
12.11	even	2		8752.2.a.s.1.17		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.10	✓	18	3.2	odd	2
4923.2.a.l.1.9		18	1.1	even	1	trivial
8752.2.a.s.1.17		18	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q+0.523506 q^{2} -1.72594 q^{4} +1.35183 q^{5} +3.60927 q^{7} -1.95055 q^{8} +O(q^{10})\)	\(q+0.523506 q^{2} -1.72594 q^{4} +1.35183 q^{5} +3.60927 q^{7} -1.95055 q^{8} +0.707691 q^{10} +0.337880 q^{11} -0.968153 q^{13} +1.88948 q^{14} +2.43075 q^{16} -1.24740 q^{17} -5.50049 q^{19} -2.33318 q^{20} +0.176882 q^{22} -3.13569 q^{23} -3.17256 q^{25} -0.506834 q^{26} -6.22939 q^{28} +7.18364 q^{29} +10.2089 q^{31} +5.17362 q^{32} -0.653022 q^{34} +4.87912 q^{35} +5.45502 q^{37} -2.87954 q^{38} -2.63682 q^{40} +7.24577 q^{41} -6.48922 q^{43} -0.583161 q^{44} -1.64155 q^{46} +7.68104 q^{47} +6.02683 q^{49} -1.66085 q^{50} +1.67097 q^{52} +4.50651 q^{53} +0.456756 q^{55} -7.04008 q^{56} +3.76068 q^{58} -2.41312 q^{59} +6.37922 q^{61} +5.34441 q^{62} -2.15308 q^{64} -1.30878 q^{65} +0.799497 q^{67} +2.15294 q^{68} +2.55425 q^{70} -2.93852 q^{71} -15.3091 q^{73} +2.85574 q^{74} +9.49353 q^{76} +1.21950 q^{77} +10.3913 q^{79} +3.28597 q^{80} +3.79321 q^{82} +13.3213 q^{83} -1.68627 q^{85} -3.39715 q^{86} -0.659054 q^{88} +4.42608 q^{89} -3.49433 q^{91} +5.41201 q^{92} +4.02107 q^{94} -7.43573 q^{95} -6.98217 q^{97} +3.15508 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

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Database

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